To determine the measure of the unknown angle [tex]\( x \)[/tex] in a right triangle where [tex]\(\sin(x) = \frac{5}{8.3}\)[/tex], we can follow these steps:
1. Understand the given values:
- The length of the side opposite the angle [tex]\( x \)[/tex] is 5 units.
- The length of the hypotenuse is 8.3 units.
2. Setup the sine function:
- By definition, [tex]\(\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}}\)[/tex].
- So, [tex]\(\sin(x) = \frac{5}{8.3}\)[/tex].
3. Calculate [tex]\( x \)[/tex]:
- To find [tex]\( x \)[/tex], take the inverse sine (arcsine) of [tex]\(\frac{5}{8.3}\)[/tex]:
[tex]\[
x = \sin^{-1}\left(\frac{5}{8.3}\right)
\][/tex]
4. Get the numerical value of [tex]\( x \)[/tex]:
- The value of [tex]\( \sin^{-1}\left(\frac{5}{8.3}\right) \)[/tex] is approximately 0.6465165714340122 radians.
5. Convert the angle to degrees:
- Since angles are often expressed in degrees, we convert radians to degrees.
- Using the conversion [tex]\(1 \text{ radian} = \frac{180}{\pi} \approx 57.2958\text{ degrees}\)[/tex]:
[tex]\[
x \text{ (in degrees)} = 0.6465165714340122 \times 57.2958 \approx 37.0426709284371 \text{ degrees}
\][/tex]
So, the measure of the unknown angle [tex]\( x \)[/tex] is approximately 0.6465 radians or 37.0427 degrees.
